A muon collider represents the ideal machine to reach very high center-of-mass energies ($\sqrt{s}=1.5-10$ TeV) and luminosities $O$($0.5-10$/ab). A large number of Higgs bosons will be produced mainly through the Vector Boson Fusion ($VBF$) processes. The $VBF$ through Z bosons ($ZZH$) production process could be difficult to disentangle from the dominant $WWZ$, since the final state $VBF$ muons, produced in the very forward region, could escape the detector. As a consequence, at a multi-TeV muon collider, the $H \rightarrow ZZ$ decay process turns out to be favoured to probe exclusively the Higgs boson coupling to Z bosons. In this paper, for the first time, a feasibility study of the search for $H \rightarrow ZZ^{*} \rightarrow 4\mu$ at a 1.5 and 3 TeV muon collider is presented. The study of the four muons final state, performed on fully simulated Monte Carlo samples, allows to optimize the muon reconstruction, thus providing feedback for the detector design. Irreducible backgrounds from Standard Model are studied. A first estimate of the senistivity of the Higgs boson coupling to Z bosons in the $4 \mu$ channel is provided, along with a preliminary evaluation of the impact of the machine background in the 1.5-TeV scenario.